Active control of wave propagation in nonlinear planar networks using piezoelectric actuation

Abstract

Due to the fascinating effects of nonlinearity on the wave propagation and dynamic properties of periodic structures, nonlinear phononic crystals have been an important research topic in recent years. However, while a lot of efforts were devoted to the wave propagation analysis of nonlinear chains, 2D planar lattices with weak nonlinearities have been left relatively unexplored. Hence, this work presents an active control technique of wave propagation in nonlinear planar lattices using piezoelectric actuation. To formulate the problem, the governing equations are derived for the transverse motion of weakly nonlinear lattices with square topology considering the effects of auxiliary piezoelectric springs. The Lindstedt–Poincare perturbation method is applied to obtain semi-analytical solutions of the governing equations. Furthermore, the analytical nonlinear dispersion relations are derived for the discrete nonlinear phononic lattices with triangular topology, for the first time. The results reveal that in addition to the effect of piezoelectric springs with both positive and negative gains on the dispersion curves, the wave directionality of nonlinear planar lattices is also affected by such modifications. Hence, a tuning approach is proposed to control the directionality and isotropy of wave propagation in nonlinear planar lattices, as well as their wave attenuation performance. According to the results, piezoelectric springs with negative control gain can open new tunable ultra-low frequency stop-bands in the dispersion curves of planar nonlinear lattices.